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Rapidly decaying Fourier-like bases.

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    Standard Fourier methods struggle with finite domains. New Fourier-like bases offer well-defined spectral moments for analyzing function variations and optical surface structures.

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    Area of Science:

    • Applied Mathematics
    • Optical Engineering
    • Signal Processing

    Background:

    • Extracting characteristic scales from function variations often relies on frequency spectrum analysis.
    • Standard Fourier methods face limitations with finite domains, failing to produce finite spectral moments.
    • This poses challenges for analyzing mid-spatial frequency structures in applications like optical surfaces.

    Purpose of the Study:

    • To develop novel Fourier-like bases capable of yielding finite spectral moments for functions defined on finite domains.
    • To address the limitations of traditional Fourier analysis in characterizing function variations.
    • To provide a suitable spectral analysis tool for investigating mid-spatial frequency structures.

    Main Methods:

    • Investigated a family of Fourier-like bases designed for rapidly decaying spectra.
    • Derived these bases by considering functions with stationary normalized mean square derivatives.
    • Applied these bases to analyze functions defined on finite domains.

    Main Results:

    • The proposed Fourier-like bases yield well-defined spectral moments even for functions on finite domains.
    • These bases exhibit rapidly decaying spectra, suitable for moment calculations.
    • The derived spectral properties align with the requirements for analyzing mid-spatial frequency structures.

    Conclusions:

    • The developed Fourier-like bases offer a robust method for spectral analysis on finite domains.
    • These bases overcome the limitations of standard Fourier methods for calculating spectral moments.
    • They provide a crucial tool for understanding mid-spatial frequency characteristics in optical systems.